Solve d ∫ f(x) wrt x=j

$$d \int f ( x ) d x = j$$

Solve for d (complex solution)

$$\left\{\begin{matrix}d=\frac{j}{\frac{fx^{2}}{2}+С}\text{, }&С\neq -\frac{fx^{2}}{2}\\d\in \mathrm{C}\text{, }&j=0\text{ and }С=-\frac{fx^{2}}{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

The equation is in standard form.

$$\left(\frac{fx^{2}}{2}+С\right)d=j$$

Divide both sides by \frac{1}{2}fx^{2}+С.

$$\frac{\left(\frac{fx^{2}}{2}+С\right)d}{\frac{fx^{2}}{2}+С}=\frac{j}{\frac{fx^{2}}{2}+С}$$

Dividing by \frac{1}{2}fx^{2}+С undoes the multiplication by \frac{1}{2}fx^{2}+С.

$$d=\frac{j}{\frac{fx^{2}}{2}+С}$$

Solve for d

$$\left\{\begin{matrix}d=\frac{j}{\frac{fx^{2}}{2}+С}\text{, }&С\neq -\frac{fx^{2}}{2}\\d\in \mathrm{R}\text{, }&j=0\text{ and }С=-\frac{fx^{2}}{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

The equation is in standard form.

$$\left(\frac{fx^{2}}{2}+С\right)d=j$$

Divide both sides by \frac{1}{2}fx^{2}+С.

$$\frac{\left(\frac{fx^{2}}{2}+С\right)d}{\frac{fx^{2}}{2}+С}=\frac{j}{\frac{fx^{2}}{2}+С}$$

Dividing by \frac{1}{2}fx^{2}+С undoes the multiplication by \frac{1}{2}fx^{2}+С.

$$d=\frac{j}{\frac{fx^{2}}{2}+С}$$

Solve for f (complex solution)

$$\left\{\begin{matrix}f=\frac{С}{x^{2}}+\frac{2j}{dx^{2}}\text{, }&d\neq 0\text{ and }x\neq 0\\f\in \mathrm{C}\text{, }&\left(j=0\text{ and }d=0\right)\text{ or }\left(j=Сd\text{ and }x=0\text{ and }d\neq 0\right)\end{matrix}\right.$$

Solve for f

$$\left\{\begin{matrix}f=\frac{С}{x^{2}}+\frac{2j}{dx^{2}}\text{, }&d\neq 0\text{ and }x\neq 0\\f\in \mathrm{R}\text{, }&\left(j=0\text{ and }d=0\right)\text{ or }\left(j=Сd\text{ and }x=0\text{ and }d\neq 0\right)\end{matrix}\right.$$